Non-Hermitian Chern Bands.

The relation between chiral edge modes and bulk Chern numbers of quantum Hall insulators is a paradigmatic example of bulk-boundary correspondence. We show that the chiral edge modes are not strictly tied to the Chern numbers defined by a non-Hermitian Bloch Hamiltonian. This breakdown of conventional bulk-boundary correspondence stems from the non-Bloch-wave behavior of eigenstates (non-Hermitian skin effect), which generates pronounced deviations of phase diagrams from the Bloch theory. We introduce non-Bloch Chern numbers that faithfully predict the numbers of chiral edge modes. The theory is backed up by the open-boundary energy spectra, dynamics, and phase diagram of representative lattice models. Our results highlight a unique feature of non-Hermitian bands and suggest a non-Bloch framework to characterize their topology.

[1]  J. González,et al.  Topological protection from exceptional points in Weyl and nodal-line semimetals , 2017, 1702.02521.

[2]  Barry C. Sanders,et al.  Observation of topological edge states in parity–time-symmetric quantum walks , 2017, Nature Physics.

[3]  M. Soljačić,et al.  Observation of bulk Fermi arc and polarization half charge from paired exceptional points , 2017, Science.

[4]  Shinsei Ryu,et al.  Classification of topological quantum matter with symmetries , 2015, 1505.03535.

[5]  Y. Ashida,et al.  Parity-time-symmetric topological superconductor , 2018, Physical Review B.

[6]  R. Aguado,et al.  Majorana bound states from exceptional points in non-topological superconductors , 2014, Scientific Reports.

[7]  Guang-Yao Huang,et al.  Topological invariance and global Berry phase in non-Hermitian systems , 2013, 1502.00443.

[8]  Y. Wang,et al.  Single-mode laser by parity-time symmetry breaking , 2014, Science.

[9]  H. Harney,et al.  PT symmetry and spontaneous symmetry breaking in a microwave billiard. , 2011, Physical review letters.

[10]  Franco Nori,et al.  Edge Modes, Degeneracies, and Topological Numbers in Non-Hermitian Systems. , 2016, Physical review letters.

[11]  Huitao Shen,et al.  Quantum Oscillation from In-Gap States and a Non-Hermitian Landau Level Problem. , 2018, Physical review letters.

[12]  R. Morandotti,et al.  Observation of PT-symmetry breaking in complex optical potentials. , 2009, Physical review letters.

[13]  M. Rudner,et al.  Topological transition in a non-Hermitian quantum walk. , 2008, Physical review letters.

[14]  Tony E. Lee,et al.  Heralded Magnetism in Non-Hermitian Atomic Systems , 2014, 1402.6700.

[15]  Pseudo-Hermiticity versus PT-symmetry. II. A complete characterization of non-Hermitian Hamiltonians with a real spectrum , 2001, math-ph/0110016.

[16]  U. Peschel,et al.  Parity–time synthetic photonic lattices , 2012, Nature.

[17]  Ling Lu,et al.  Spawning rings of exceptional points out of Dirac cones , 2015, Nature.

[18]  B. Bernevig.  Topological Insulators and Topological Superconductors , 2013 .

[19]  D. Christodoulides,et al.  Parity-time–symmetric microring lasers , 2014, Science.

[20]  Hui Cao,et al.  Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics , 2015 .

[21]  Lan Yang,et al.  Exceptional points enhance sensing in an optical microcavity , 2017, Nature.

[22]  S. Fan,et al.  Effects of non-Hermitian perturbations on Weyl Hamiltonians with arbitrary topological charges , 2017, 2018 Conference on Lasers and Electro-Optics (CLEO).

[23]  Liang Fu,et al.  Topological Band Theory for Non-Hermitian Hamiltonians. , 2017, Physical review letters.

[24]  G. Weick,et al.  Topological collective plasmons in bipartite chains of metallic nanoparticles , 2016, 1611.03349.

[25]  D. Thouless,et al.  Quantized Hall conductance in a two-dimensional periodic potential , 1992 .

[26]  M Segev,et al.  Topologically protected bound states in photonic parity-time-symmetric crystals. , 2017, Nature materials.

[27]  Tony E. Lee,et al.  Anomalous Edge State in a Non-Hermitian Lattice. , 2016, Physical review letters.

[28]  Zhong Wang,et al.  Edge States and Topological Invariants of Non-Hermitian Systems. , 2018, Physical review letters.

[29]  Haldane,et al.  Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the "parity anomaly" , 1988, Physical review letters.

[30]  A. Mostafazadeh.  Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian , 2001, math-ph/0107001.

[31]  J. Gong,et al.  Geometric phase in PT-symmetric quantum mechanics , 2010, 1003.3076.

[32]  Kevin Barraclough,et al.  I and i , 2001, BMJ : British Medical Journal.

[33]  Z. Q. Zhang,et al.  The emergence, coalescence and topological properties of multiple exceptional points and their experimental realization , 2015, 1509.06886.

[34]  T. D. Lee,et al.  Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model , 1952 .

[35]  Demetrios N. Christodoulides,et al.  Enhanced sensitivity at higher-order exceptional points , 2017, Nature.

[36]  Shiyue Hua,et al.  Parity–time symmetry and variable optical isolation in active–passive-coupled microresonators , 2014, Nature Photonics.

[37]  Hui Cao,et al.  Unidirectional invisibility induced by PT-symmetric periodic structures. , 2011, Physical review letters.

[38]  X. Qi,et al.  Topological insulators and superconductors , 2010, 1008.2026.

[39]  Carmichael,et al.  Quantum trajectory theory for cascaded open systems. , 1993, Physical review letters.

[40]  Luyao Jiang,et al.  Topological energy transfer in an optomechanical system with exceptional points , 2016, Nature.

[41]  R. Lu,et al.  Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-Hermitian systems , 2018, 1802.04169.

[42]  Ulrich Kuhl,et al.  Selective enhancement of topologically induced interface states in a dielectric resonator chain , 2014, Nature Communications.

[43]  L. Duan,et al.  Weyl Exceptional Rings in a Three-Dimensional Dissipative Cold Atomic Gas. , 2016, Physical review letters.

[44]  Xiao-Liang Qi,et al.  Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors , 2005, cond-mat/0505308.

[45]  M. Bandres,et al.  Complex Edge-State Phase Transitions in 1D Topological Laser Arrays , 2017, 2018 Conference on Lasers and Electro-Optics (CLEO).

[46]  Luis E. F. Foa Torres,et al.  Non-Hermitian robust edge states in one dimension: Anomalous localization and eigenspace condensation at exceptional points , 2017, 1711.05235.

[47]  Hua Long,et al.  Topological edge modes in non-Hermitian plasmonic waveguide arrays. , 2017, Optics express.

[48]  S. Longhi,et al.  Bloch oscillations in complex crystals with PT symmetry. , 2009, Physical review letters.

[49]  P. Zoller,et al.  Topology by dissipation in atomic quantum wires , 2011, 1105.5947.

[50]  Vilson R. Almeida,et al.  Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies. , 2013, Nature materials.

[51]  M. Bandres,et al.  Topological insulator laser: Experiments , 2018, Science.

[52]  Stefan Nolte,et al.  Observation of a Topological Transition in the Bulk of a Non-Hermitian System. , 2015, Physical review letters.

[53]  M. Soljačić,et al.  Topological photonics , 2014, Nature Photonics.

[54]  Andrea Alù,et al.  An invisible acoustic sensor based on parity-time symmetry , 2015, Nature Communications.

[55]  Masahito Ueda,et al.  Zeno Hall Effect. , 2016, Physical review letters.

[56]  Ingrid Rotter,et al.  A non-Hermitian Hamilton operator and the physics of open quantum systems , 2009 .

[57]  H. Menke,et al.  Topological quantum wires with balanced gain and loss , 2017, 1701.09009.

[58]  장윤희,et al.  Y. , 2003, Industrial and Labor Relations Terms.

[59]  Li Ge,et al.  Pump-induced exceptional points in lasers. , 2011, Physical review letters.

[60]  Henning Schomerus,et al.  Topologically Protected Defect States in Open Photonic Systems with Non-Hermitian Charge-Conjugation and Parity-Time Symmetry. , 2015, Physical review letters.

[61]  T. Hughes,et al.  Absence of topological insulator phases in non-Hermitian PT-symmetric Hamiltonians , 2011, 1107.1064.

[62]  M. Kohmoto,et al.  Edge states and topological phases in non-Hermitian systems , 2011, 1107.2079.

[63]  Demetrios N. Christodoulides,et al.  Non-Hermitian physics and PT symmetry , 2018, Nature Physics.

[64]  C. Bender,et al.  Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry , 1997, physics/9712001.

[65]  Q. Xue,et al.  Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator , 2013, Science.

[66]  R. Lu,et al.  PT symmetry in the non-Hermitian Su-Schrieffer-Heeger model with complex boundary potentials , 2014, 1405.5591.

[67]  Han Zhao,et al.  Topological hybrid silicon microlasers , 2017, 2018 Conference on Lasers and Electro-Optics (CLEO).

[68]  R. Sarpong,et al.  Bio-inspired synthesis of xishacorenes A, B, and C, and a new congener from fuscol† †Electronic supplementary information (ESI) available. See DOI: 10.1039/c9sc02572c , 2019, Chemical science.

[69]  Topological Photonics , 2014, 1408.6730.

[70]  M. Bandres,et al.  Topological insulator laser: Theory , 2018, Science.

[71]  Gorjan Alagic,et al.  #p , 2019, Quantum information & computation.

[72]  M. Bandres,et al.  Complex Edge-State Phase Transitions in 1D Topological Laser Arrays , 2017, 2018 Conference on Lasers and Electro-Optics (CLEO).

[73]  M. Segev,et al.  Observation of parity–time symmetry in optics , 2010 .

[74]  Tony E. Lee,et al.  Entanglement and spin squeezing in non-Hermitian phase transitions. , 2014, Physical review letters.

[75]  J. Main,et al.  Relation between PT -symmetry breaking and topologically nontrivial phases in the Su-Schrieffer-Heeger and Kitaev models , 2017, 1702.00173.

[76]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[77]  S. Lieu.  Topological phases in the non-Hermitian Su-Schrieffer-Heeger model , 2017, 1709.03788.

[78]  P. Tong,et al.  Spontaneous PT -symmetry breaking in non-Hermitian Kitaev and extended Kitaev models , 2015 .

[79]  C. Kane,et al.  Topological Insulators , 2019, Electromagnetic Anisotropy and Bianisotropy.

[80]  J. Gong,et al.  Dynamical quantum phase transitions in non-Hermitian lattices , 2017, Physical Review A.

[81]  Z. Musslimani,et al.  Beam dynamics in PT symmetric optical lattices. , 2008, Physical review letters.

[82]  C. Schneider,et al.  Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard , 2015, Nature.

[83]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[84]  Shachar Klaiman,et al.  Visualization of branch points in PT-symmetric waveguides. , 2008, Physical review letters.

[85]  P. Xue,et al.  Detecting Topological Invariants in Nonunitary Discrete-Time Quantum Walks. , 2017, Physical review letters.

[86]  C. Yuce.  Majorana edge modes with gain and loss , 2016, 1605.09597.

[87]  Masahito Ueda,et al.  Information Retrieval and Criticality in Parity-Time-Symmetric Systems. , 2017, Physical review letters.

[88]  A. Zyuzin,et al.  Flat band in disorder-driven non-Hermitian Weyl semimetals , 2017, 1710.05344.

[89]  Zheng Wang,et al.  Observation of unidirectional backscattering-immune topological electromagnetic states , 2009, Nature.

[90]  Z. Song,et al.  Topological phases in a Kitaev chain with imbalanced pairing , 2017, 1707.04718.

[91]  H. Yilmaz,et al.  Loss-induced suppression and revival of lasing , 2014, Science.

[92]  L. Christophorou.  Science , 2018, Emerging Dynamics: Science, Energy, Society and Values.

[93]  Y. Chong,et al.  Exceptional points in a non-Hermitian topological pump , 2017, 1703.01293.

[94]  Sungsam Kang,et al.  Quasieigenstate coalescence in an atom-cavity quantum composite. , 2010, Physical review letters.

[95]  Y. Ashida,et al.  Parity-time-symmetric quantum critical phenomena , 2016, Nature Communications.

[96]  Carl M. Bender,et al.  Making sense of non-Hermitian Hamiltonians , 2007, hep-th/0703096.